Physical properties of foods

In the first chap-ter physical attributes of foods which are size, shape, volume, density and porosity are discussed.Methods to measure these properties are explained in details.. Size, S

of Foods

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Titles Elementary Food Science, Fourth Edition, Ernest R Vieira (1996)

Essentials of Food Sanitation, Norman G Marriott (1997)

Essentials of Food Science, Second Edition, Vickie A Vaclavik and Elizabeth W Christian (2003)

Food Analysis, Third Edition, S Suzanne Nielsen (2003)

Food Analysis Laboratory Manual, S Suzanne Nielsen (2003)

Food Science, Fifth Edition, Norman N Potter and Joseph H Hotchkiss (1995)

Fundamentals of Food Processing, Third Edition, Romeo T Toledo (2006)

Introduction to Food Processing, P G Smith (2003)

Modern Food Microbiology, Seventh Edition, James M Jay, Martin J Loessner,

and David A Golden (2005)

Physical Properties of Foods, Serpil Sahin and Servet G¨ul¨um Sumnu (2006)

Principles of Food Chemistry, Third Edition, John M de Man (1999)

Principles of Food Processing, Dennis R Heldman and Richard W Hartel (1997)

Principles of Food Sanitation, Fifth Edition, Norman G Marriott and Robert B Gravani (2006)

Sensory Evaluation of Food: Principles and Practices, Harry T Lawless and Hildegarde Heymann (1998)

Physical Properties

of Foods

Serpil Sahin and Servet G¨ul¨um Sumnu

Middle East Technical University

Ankara, Turkey

Serpil Sahin

Department of Food Engineering

Middle East Technical University

Turkeygulum@metu.edu.tr

Library of Congress Control Number: 2005937128

ISBN-10: 0-387-30780-X e-ISBN 0-387-30808-3 Printed on acid-free paper.

ISBN-13: 978-0387-30780-0

C

2006 Springer Science+Business Media, LLC.

All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts

in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such,

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SEM˙IHA-S¸EVKET SAHIN

&

Who have given us our roots

We tried to write a book to provide a fundamental understanding of physical properties of foods.

In this book, the knowledge of physical properties is combined with food science, physical chemistry,physics and engineering knowledge Physical properties data are required during harvesting, process-ing, storage and even shipping to the consumer The material in the book will be helpful for the students

to understand the relationship between physical and functional properties of raw, semi-finished andprocessed food to obtain products with desired shelf-life and quality

This book discusses basic definitions and principles of physical properties, the importance of ical properties in food industry and measurement methods Moreover, recent studies in the area ofphysical properties are summarized In addition, each chapter is provided with examples and problems.These problems will be helpful for students for their self-study and to gain information how to analyzeexperimental data to generate practical information

phys-This book is written to be a textbook for undergraduate students which will fill the gap in physicalproperties area In addition, the material in the book may be of interest to people who are working in thefield of Food Science, Food Technology, Biological Systems Engineering, Food Process Engineeringand Agricultural Engineering It will also be helpful for graduate students who deal with physicalproperties in their research

The physical properties of food materials are discussed in 6 main categories such as size, shape,volume and related physical attributes, rheological properties, thermal properties, electromagneticproperties, water activity and sorption properties and surface properties in this book In the first chap-ter physical attributes of foods which are size, shape, volume, density and porosity are discussed.Methods to measure these properties are explained in details In Chapter 2, after making an intro-duction on Newtonian and non-Newtonian fluid flow, viscosity measurement methods are discussed.Then, principle of viscoelastic fluids, methods to determine the viscoelastic behavior and modelsused in viscoelastic fluids are mentioned Chapter 3 explains definition and measurement methods ofthermal properties such as thermal conductivity, specific heat, thermal diffusivity and enthalpy In the

vii

fourth chapter color and dielectric properties of foods are covered In Chapter 5, equilibrium criteriaand colligative properties are discussed Then, information is given on measurement of water activity.Finally preparation of sorption isotherms and models are discussed The last chapter is about surfaceproperties and their measurement methods Where appropriate, we have cited throughout the text thearticles that are available for more information.

We are deeply grateful to Prof Dr Haluk Hamamci for encouraging us during writing this bookand his belief in us We would also like to thank our colleagues Prof Dr Ali Esin, Prof Dr HalukHamamci, Assoc Prof Dr Nihal Aydogan, Assoc Prof Dr Pinar Calik, Assoc Prof Dr NaimeAsli Sezgi, Assoc Prof Dr Esra Yener, Assist Prof Dr Yusuf Uludag, who read the chapters andgave useful suggestions We would also like to thank our Ph.D student Halil Mecit Oztop and hisbrother Muin S Oztop for drawing some of the figures We would like to extend our thanks to ourPh.D students, Isil Barutcu, Suzan Tireki, Semin Ozge Keskin, Elif Turabi and Nadide Seyhun forreviewing our book We are happy to acknowledge the teaching assistants Aysem Batur and IncinurHasbay for their great effort in drawing some of the figures, finding the examples and problems given

in each chapter

Last but not the least; we would like to thank our families for their continuous support throughoutour academic career With love, this work is dedicated to our parents who have patience and belief in

us Thank you for teaching us how to struggle the difficulties in life

1—Size, Shape, Volume, and Related Physical Attributes 1

Summary 1

1.1 Size 1

1.2 Shape 3

1.3 Particle Size Distribution 8

1.4 Volume 15

1.4.1 Liquid Displacement Method 16

1.4.2 Gas Displacement Method 18

1.4.3 Solid Displacement Method 19

1.4.4 Expressions of Volume 20

1.5 Density 20

1.6 Porosity 24

1.7 Determination of Volume of Different Kinds of Pores 30

1.8 Shrinkage 33

Problems 35

References 36

2—Rheological Properties of Foods 39

Summary 39

2.1 Introduction to Rheology 39

2.2 Flow of Material 40

2.2.1 Newton’s Law of Viscosity 40

2.2.2 Viscous Fluids 44

2.2.2.1 Newtonian Fluids 44

2.2.2.2 Non-Newtonian Fluids 44

2.2.3 Plastic Fluids 47

2.2.3.1 Bingham Plastic Fluids 47

2.2.3.2 Non-Bingham Plastic Fluids 47

2.2.4 Time Dependency 47

2.2.5 Solution Viscosity 49

ix

2.3 Viscosity Measurement 50

2.3.1 Capillary Flow Viscometers 50

2.3.2 Orifice Type Viscometers 59

2.3.3 Falling Ball Viscometers 59

2.3.4 Rotational Viscometers 61

2.3.4.1 Concentric Cylinder (Coaxial Rotational) Viscometers 61

2.3.4.2 Cone and Plate Viscometers 64

2.3.4.3 Parallel Plate Viscometers 68

2.3.4.4 Single-Spindle Viscometers (Brookfield Viscometer) 69

2.3.5 Other Types of Viscometers 70

2.3.5.1 Vibrational (Oscillation) Viscometer 70

2.3.5.2 Bostwick Consistometer 70

2.4 Deformation of Material 71

2.5 Viscoelastic Behavior 75

2.5.1 Stress Relaxation Test 76

2.5.2 Creep Test 77

2.5.3 Dynamic Test (Oscillatory Test) 78

2.6 Extensional Flow 79

2.7 Mechanical Models 80

2.7.1 Elastic (Spring) Model 80

2.7.2 Viscous (Dashpot) Model 81

2.7.3 Combination Models 82

2.7.3.1 Maxwell Model 82

2.7.3.2 Kelvin-Voigt Model 84

2.7.3.3 Burger Model 85

2.8 Texture of Foods 90

2.8.1 Compression 90

2.8.2 Snapping-Bending 90

2.8.3 Cutting Shear 92

2.8.4 Puncture 92

2.8.5 Penetration 93

2.8.6 Texture Profile Analysis 93

2.9 Dough Testing Instruments 96

2.9.1 Farinograph and Mixograph 96

2.9.2 Extensograph and Alveograph 98

2.9.3 Amylograph 100

Problems 100

References 104

3—Thermal Properties of Foods 107

Summary 107

3.1 Fourier’s Law of Heat Conduction 107

3.2 Thermal Conductivity 109

3.2.1 Prediction of Thermal Conductivity 112

3.2.1.1 Parallel Model 113

3.2.1.2 Series (Perpendicular) Model 114

Contents xi

3.2.1.3 Krischer Model 114

3.2.1.4 Maxwell-Eucken Model 114

3.2.1.5 Kopelman Model 115

3.2.1.6 Improved Thermal Conductivity Prediction Models 116

3.2.2 Measurement of Thermal Conductivity 120

3.2.2.1 Steady State Methods 121

3.2.2.2 Unsteady-State Methods 125

3.3 Specific Heat 139

3.3.1 Prediction of Specific Heat 139

3.3.2 Measurement of Specific Heat 142

3.3.2.1 Method of Mixture 142

3.3.2.2 Method of Guarded Plate 143

3.3.2.3 Method of Comparison Calorimeter 143

3.3.2.4 Adiabatic Agricultural Calorimeter 145

3.3.2.5 Differential Scanning Calorimeter (DSC) 145

3.3.2.6 Method of Calculated Specific Heat 148

3.4 Enthalpy and Latent Heat 148

3.5 Thermal Diffusivity 149

3.5.1 Indirect Prediction Method 149

3.5.2 Direct Measurement Methods 150

3.5.2.1 The Temperature History Method 150

3.5.2.2 Thermal Conductivity Probe 150

3.5.2.3 Dickerson Method 150

Problems 151

References 153

4—Electromagnetic Properties 157

Summary 157

4.1 Interaction of Objects with Light 157

4.2 Color 162

4.2.1 Color Measuring Equipments 164

4.2.1.1 Spectrophotometers 164

4.2.1.2 Colorimeters 165

4.2.2 Color Order Systems 166

4.2.2.1 Munsell Color System 166

4.2.2.2 CIE Color System 167

4.2.2.3 CIE L∗a∗b∗(CIELAB) Color Spaces 167

4.2.2.4 Hunter Lab Color Space 169

4.2.2.5 Lovibond System 169

4.2.3 Color Differences 170

4.3 Dielectric Properties of Foods 173

4.3.1 Basic Principles of Microwave Heating 173

4.3.1.1 Ionic Interaction (Ionic Conduction) 173

4.3.1.2 Dipolar Rotation 173

4.3.2 Definition of Dielectric Properties 174

4.3.3 Effects of Moisture Content on Dielectric Properties 177

4.3.4 Effects of Temperature on Dielectric Properties 178

4.3.5 Effects of Composition of Foods on Dielectric Properties 180

4.3.5.1 Dielectric Properties of Salt Solutions 181

4.3.5.2 Dielectric Properties of Carbohydrates 181

4.3.5.3 Dielectric Properties of Proteins 185

4.3.5.4 Dielectric Properties of Fat 186

4.3.6 Assessment of Quality of Foods by Using Dielectric Properties 186

4.3.7 Measurement of Dielectric Properties 187

References 189

5—Water Activity and Sorption Properties of Foods 193

Summary 193

5.1 Criteria of Equilibrium 193

5.2 Ideal Solution—Raoult’s Law 196

5.3 Henry’s Law 197

5.4 Colligative Properties 197

5.4.1 Boiling Point Elevation 197

5.4.2 Freezing Point Depression 200

5.4.3 Osmotic Pressure 203

5.5 Equilibria in Nonideal Systems—Fugacity and Activity 204

5.6 Water Activity 205

5.7 Prediction of Water Activity 206

5.8 Water Activity Measurement Methods 209

5.8.1 Measurements Based on Colligative Properties 210

5.8.1.1 Water Activity Determination by Vapor Pressure Measurement 210

5.8.1.2 Water Activity Determination by Freezing Point Depression 211

5.8.2 Measurements Based on Isopiestic Transfer 211

5.8.3 Measurements Using Hygrometers 211

5.8.4 Measurements Based on Hygroscopicity of Salts 212

5.9 Effects of Temperature on Water Activity 212

5.10 Effects of Pressure on Water Activity 212

5.11 Adjustment of Water Activity and Preparation of Moisture Sorption Isotherms 213

5.11.1 Hysteresis 216

5.11.2 Isotherm Models 218

Problems 224

References 226

6—Surface Properties of Foods 229

Summary 229

6.1 Surface Tension 229

6.2 Laplace Equation 232

6.3 Kelvin Equation 234

6.4 Surface Activity 235

6.5 Interfacial Tension 237

6.6 Young and Dupre’ Equations 237

Contents xiii

6.7 Colloidal Systems in Foods 239

6.7.1 Sols 239

6.7.2 Gels 240

6.7.3 Emulsions 240

6.7.4 Foams 243

6.8 Measurement of Contact Angle and Surface Tension 244

6.8.1 Contact Angle Measurement Methods 244

6.8.2 Surface Tension Measurement Methods 245

Problems 248

References 248

Index 251

Size, Shape, Volume, and Related

Size is an important physical attribute of foods used in screening solids to separate foreign materials,grading of fruits and vegetables, and evaluating the quality of food materials In fluid flow, and heatand mass transfer calculations, it is necessary to know the size of the sample Size of the particulatefoods is also critical For example, particle size of powdered milk must be large enough to preventagglomeration, but small enough to allow rapid dissolution during reconstitution Particle size wasfound to be inversely proportional to dispersion of powder and water holding capacity of whey proteinpowders (Resch & Daubert, 2001) Decrease in particle size also increased the steady shear and

1

2 1 Size, Shape, Volume, and Related Physical Attributes

Figure 1.1 Micrometer

complex viscosity of the reconstituted powder The powder exhibited greater intrinsic viscosity asthe particle size increased The size of semolina particles was found to influence mainly sorptionkinetics (Hebrard, Oulahna, Galet, Cuq, Abecassis, & Fages, 2003) The importance of particle sizemeasurement has been widely recognized, especially in the beverage industry, as the distribution andconcentration ratio of particulates present in beverages greatly affect their flavor

It is easy to specify size for regular particles, but for irregular particles the term size must bearbitrarily specified

Particle sizes are expressed in different units depending on the size range involved Coarse particlesare measured in millimeters, fine particles in terms of screen size, and very fine particles in micrometers

or nanometers Ultrafine particles are sometimes described in terms of their surface area per unit mass,usually in square meters per gram (McCabe, Smith & Harriot, 1993)

Size can be determined using the projected area method In this method, three characteristic sions are defined:

dimen-1 Major diameter, which is the longest dimension of the maximum projected area;

2 Intermediate diameter, which is the minimum diameter of the maximum projected area or themaximum diameter of the minimum projected area; and

3 Minor diameter, which is the shortest dimension of the minimum projected area

Length, width, and thickness terms are commonly used that correspond to major, intermediate, andminor diameters, respectively

The dimensions can be measured using a micrometer or caliper (Fig 1.1) The micrometer is asimple instrument used to measure distances between surfaces Most micrometers have a frame,

anvil, spindle, sleeve, thimble, and ratchet stop They are used to measure the outside diameters, insidediameters, the distance between parallel surfaces, and the depth of holes.

Particle size of particulate foods can be determined by sieve analysis, passage through an electricallycharged orifice, and settling rate methods Particle size distribution analyzers, which determine boththe size of particles and their state of distribution, are used for production control of powders

Shape is also important in heat and mass transfer calculations, screening solids to separate foreignmaterials, grading of fruits and vegetables, and evaluating the quality of food materials The shape of

a food material is usually expressed in terms of its sphericity and aspect ratio

Sphericity is an important parameter used in fluid flow and heat and mass transfer calculations.

Sphericity or shape factor can be defined in different ways

According to the most commonly used definition, sphericity is the ratio of volume of solid tothe volume of a sphere that has a diameter equal to the major diameter of the object so that it

can circumscribe the solid sample For a spherical particle of diameter D p, sphericity is equal to 1(Mohsenin, 1970)

= sphericity,

V e= volume of the triaxial ellipsoid with equivalent diameters (m3),

V c= volume of the circumscribed sphere (m3)

In a triaxial ellipsoid, all three perpendicular sections are ellipses (Fig 1.2) If the major,

interme-diate, and minor diameters are 2a, 2b, and 2c, respectively, volume of the triaxial ellipsoid can be

determined from the following equation:

4 1 Size, Shape, Volume, and Related Physical Attributes

b

b a

c

Figure 1.2 Triaxial ellipsoid

where

D p= equivalent diameter or nominal diameter of the particle (m),

S p= surface area of one particle (m2),

V p = volume of one particle (m3)

The equivalent diameter is sometimes defined as the diameter of a sphere having the same volume

as the particle However, for fine granular materials, it is difficult to determine the exact volume andsurface area of a particle Therefore, equivalent diameter is usually taken to be the nominal size based

on screen analysis or microscopic examination in granular materials The surface area is found fromadsorption measurements or from the pressure drop in a bed of particles

In general, diameters may be specified for any equidimensional particle Particles that are notequidimensional, that is, longer in one direction than in others, are often characterized by the secondlongest major dimension For example, for needlelike particles, equivalent diameter refers to thethickness of the particles, not their length

In a sample of uniform particles of diameter D p, the number of particles in the sample is:

p V p

(1.6)where

N = the number of particles,

m= mass of the sample (kg),

ρ p = density of the particles (kg/m3),

V p = volume of one particle (m3)

Total surface area of the particles is obtained from Eqs (1.5) and (1.6):

A = N S p=
ρ 6m

Another definition of sphericity is the ratio of the diameter of the largest inscribed circle (d i) to the

diameter of the smallest circumscribed circle (d c) (Mohsenin, 1970):

d c

(1.8)Recently, Bayram (2005) proposed another equation to calculate sphericity as:

D i = any measured dimension (m),

D= average dimension or equivalent diameter (m),

N = number of measurements (the increase in N increases the accuracy).

According to this formula, equivalent diameter for irregular shape material is accepted as the averagedimension Differences between average diameter and measured dimensions are determined by thesum of square of differences When this difference is divided by the square of product of the averagediameter and number of measurements, it gives a fraction for the approach of the slope to an equivalentsphere, which is sphericity

According to Eq (1.9), if the sample sphericity value is close to zero it can be ered as spherical Table 1.1 shows sphericity values of some granular materials determined by

From Bayram (2005).

6 1 Size, Shape, Volume, and Related Physical Attributes

The radius of the sphere (r s) having this volume can be calculated as:

The aspect ratio (Ra) is another term used to express the shape of a material It is calculated using

the length (a) and the width (b) of the sample as (Maduako & Faborode, 1990):

Certain parameters are important for the design of conveyors for particulate foods, such as radius of

curvature, roundness, and angle of repose Radius of curvature is important to determine how easily

the object will roll The more sharply rounded the surface of contact, the greater will be the stressesdeveloped A simple device for measuring the radius of curvature is shown in Fig 1.3 It consists of ametal base that has a dial indicator and holes into which pins are placed Two pins are placed withinthese holes according to the size of the object When the two pins make contact with the surface,

the tip of the dial indicator is pushed upwards Then, the dial indicator reads the sagittal height (S).

The radius of curvature is calculated from the measured distances using this simple device and thefollowing formula:

S D

Figure 1.3 A device for measuring the radius of curvature

where

Rmin= Minimum radius of curvature (m),

Rmax= Maximum radius of curvature (m),

H= intermediate diameter or the average of minor and major diameters (m),

L = major diameter (m)

Example 1.2 The major diameter (L) and the average of the minor and major diameters (H ) of

barley are measured as 8.76 mm and 2.83 mm, respectively Calculate the minimum and maximumradii of curvature for the barley

Roundness is a measure of the sharpness of the corners of the solid Several methods are available

for estimating roundness The most commonly used ones are given below (Mohsenin, 1970):Roundness= Ap

8 1 Size, Shape, Volume, and Related Physical Attributes

Ap= largest projected area of object in natural rest position (m2),

Ac= Area of the smallest circumscribing circle as defined in Fig 1.4a (m2)

Roundness can also be estimated from Eq (1.15):

r = radius of curvature as defined in Fig 1.4b (m),

R= radius of the maximum inscribed circle (m),

N = total number of corners summed in numerator

Angle of repose is another important physical property used in particulate foods such as seeds,

grains, and fruits When granular solids are piled on a flat surface, the sides of the pile are at a definitereproducible angle with the horizontal This angle is called the angle of repose of the material Theangle of repose is important for the design of processing, storage, and conveying systems of particulatematerial When the grains are smooth and rounded, the angle of repose is low For very fine and stickymaterials the angle of repose is high For determination of this property, a box with open sides atthe top and bottom is placed on a surface The angle of repose is determined by filling the box withsample and lifting up the box gradually, allowing the sample to accumulate and form a conical heap

on the surface Then, the angle of repose is calculated from the ratio of the height to the base radius

of the heap formed

The range of particle size in foods depends on the cell structure and the degree of processing.The hardness of grain is a significant factor in the particle size distribution of flour The particle size

distribution of flour is known to play an important role in its functional properties and the quality

of end products The relationship between the physicochemical properties of rice grains and particlesize distributions of rice flours from different rice cultivars were examined (Chen, Lii, & Lu, 2004) Itwas found that physical characteristics of rice grain were the major factors but chemical compositionswere also important in affecting the particle size distribution of rice flour

To apply Eqs (1.6) and (1.7) to a mixture of particles having various sizes and densities, the mixture

is sorted into fractions, each of constant density and approximately constant size Each fraction can

be weighed or the individual particles in it can be counted Then, Eqs (1.6) and (1.7) can be applied

to each fraction and the results can be added

Particles can be separated into fractions by using one of the following methods:

1 Air elutriation method: In this method, the velocity of an air stream is adjusted so that particlesmeasuring less than a given diameter are suspended After the particles within the size range arecollected, the air velocity is increased and the new fraction of particles is collected The processcontinues until the particulate food is separated into different fractions

2 Settling, sedimentation, and centrifugation method: In settling and sedimentation, the particlesare separated from the fluid by gravitational forces acting on the particles The particles can besolid particles or liquid drops Settling and sedimentation are used to remove the particles fromthe fluid It is also possible to separate the particles into fractions of different size or density.Particles that will not settle by gravitational force can be separated by centrifugal force If thepurpose is to separate the particles into fractions of different sizes, particles of uniform densitybut different sizes are suspended in a liquid and settle at different rates Particles that settle ingiven time intervals are collected and weighed

3 Screening: This is a unit operation in which various sizes of solid particles are separated intotwo or more fractions by passing over screen(s) A dispersing agent may be added to improvesieving characteristics Screen is the surface containing a number of equally sized openings Theopenings are square Each screen is identified in meshes per inch Mesh is defined as open spaces

in a network The smallest mesh means largest clear opening

A set of standard screens is stacked one upon the other with the smallest opening at the bottomand the largest at the top placed on an automatic shaker for screen analysis (sieve analysis) In screenanalysis, the sample is placed on the top screen and the stack is shaken mechanically for a definitetime The particles retained on each screen are removed and weighed Then, the mass fractions ofparticles separated are calculated Any particles that pass through the finest screen are collected in apan at the bottom of the stack

Among the standard screens, the Tyler Standard Screen Series is the most commonly used sieveseries (Table 1.2) The area of openings in any screen in the series is exactly twice the openings inthe next smaller screen The ratio of actual mesh dimension of any screen to that of the next smallerscreen is√

2= 1.41 For closer sizing, intermediate screens are available that have mesh dimension

4

√

2= 1.189 times that of the next smaller standard screen.

Since the particles on any one screen are passed by the screen immediately ahead of it, two numbersare required to specify the size range of an increment: one for the screen through which the fractionpasses and the other on which it is retained For example, 6/8 refers to the particles passing through

the 6-mesh and remaining on an 8-mesh screen

Particle size analysis can be done in two different ways: differential analysis and cumulative analysis

In differential analysis, mass or number fraction in each size increment is plotted as a function ofaverage particle size or particle size range The results are often presented as a histogram as shown inFig 1.5 with a continuous curve to approximate the distribution If the particle size ranges are all equal

10 1 Size, Shape, Volume, and Related Physical Attributes

Table 1.2 Tyler Standard Screen Scale

21 2 a

0.312 7.925

3 0.263 6.68

31 2 a

2 instead of 1: √

2.

as in this figure, the data can be plotted directly However, it gives a false impression if the coveredrange of particle sizes differs from increment to increment Less material is retained in an incrementwhen the particle size range is narrow than when it is wide Therefore, average particle size or size

range versus X w i

D pi+1− D pi should be plotted, where X w

i is the mass fraction and

D pi+1− D pi



is the particle size range in increment i (McCabe et al., 1993).

Cumulative analysis is obtained by adding, consecutively, the individual increments, starting withthat containing the smallest particles and plotting the cumulative sums against the maximum particlediameter in the increment In a cumulative analysis, the data may appropriately be represented by

a continuous curve Table 1.3 shows a typical screen analysis Cumulative plots are made using thesecond and fifth columns of Table 1.3 (Fig 1.6)

Figure 1.5 Particle size distributions using differential analysis.

Calculations of average particle size, specific surface area, or particle population of a mixture may

be based on either a differential or a cumulative analysis In cumulative analysis, the assumption of

“all particles in a single fraction are equal in size” is not required Therefore, methods based on thecumulative analysis are more precise than those based on differential analysis

Table 1.3 A Typical Screen Analysis

12 1 Size, Shape, Volume, and Related Physical Attributes

Figure 1.6 Particle size distributions using cumulative analysis

If the particle density and sphericity are known, the surface area of the particles in each fractionmay be calculated from Eq (1.7) and the results for each fraction are added to give the specific surfacearea of mixture The specific surface area is defined as the total surface area of a unit mass of particles.For constant density (ρ p) and sphericity (
), specific surface area (A w) of the mixture is:

i= subscript showing individual increments,

X i w= mass fraction in a given increment,

n= number of increments,

D pi = average particle diameter taken as the arithmetic mean of the smallest and largest particle

diameters in the increment (m) and expressed as

of particles in each fraction is known For differential analysis:

N i= number of particles in each fraction,

N T = total number of particles,

n= number of size groups

For cumulative analysis:

14 1 Size, Shape, Volume, and Related Physical Attributes

whereϕ is the volume shape factor, which is defined by the ratio of volume of a particle (V p) to itscubic diameter:

D3

p

(1.27)Dividing the total volume of the sample by the number of particles in the mixture gives the averagevolume of a particle The diameter of such a particle is the volume mean diameter, which is found from:

Example 1.3 Wheat flour is made by grinding the dry wheat grains Particle size is an important

characteristic in many of the wheat products For example, in making wafers, if the flour is too fine,light and tender products are formed On the other hand, incomplete sheets of unsatisfactory wafersare formed if the flour is too coarse Therefore, it is important to test the grinding performance of flour

by sieve analysis in wafer producing factories Determine the volume surface mean diameter, massmean diameter, and volume mean diameter of wheat flour by differential analysis using the data given

Table E.1.3.2 Differential Analysis

of Wheat Flour

Mass Fraction

8/10 0.080 2.00710/20 0.475 1.24220/32 0.183 0.66432/42 0.049 0.42342/60 0.099 0.29960/80 0.026 0.21180/100 0.028 0.161100/Pan 0.060 0.074

Volume is defined as the amount of three-dimensional space occupied by an object, usually expressed

in units that are the cubes of length units, such as cubic inches and cubic centimeters, or in units ofliquid measure, such as gallons and liters In the SI system, the unit of volume is m3

16 1 Size, Shape, Volume, and Related Physical Attributes

Volume is an important quality attribute in the food industry It appeals to the eye, and is related toother quality parameters For instance, it is inversely correlated with texture

Volume of solids can be determined by using the following methods:

1 Volume can be calculated from the characteristic dimensions in the case of objects with regularshape

2 Volumes of solids can be determined experimentally by liquid, gas, or solid displacementmethods

3 Volume can be measured by the image processing method An image processing method hasbeen recently developed to measure volume of ellipsoidal agricultural products such as eggs,lemons, limes, and peaches (Sabliov, Boldor, Keener, & Farkas, 2002)

Liquid, gas, and solid displacement methods are described in the following sections

1.4.1 Liquid Displacement Method

If the solid sample does not absorb liquid very fast, the liquid displacement method can be used

to measure its volume In this method, volume of food materials can be measured by pycnometers(specific gravity bottles) or graduated cylinders The pycnometer has a small hole in the lid that allowsliquid to escape as the lid is fitted into the neck of the bottle (Fig 1.7) The bottle is precisely weighedand filled with a liquid of known density The lid is placed on the bottle so that the liquid is forcedout of the capillary Liquid that has been forced out of the capillary is wiped from the bottle and thebottle is weighed again After the bottle is emptied and dried, solid particles are placed in the bottleand the bottle is weighed again The bottle is completely filled with liquid so that liquid is forced fromthe hole when the lid is replaced The bottle is reweighed and the volume of solid particles can be

Figure 1.7 Pycnometer (specific gravity bottle)

determined from the following formula:

V s = Weight of the liquid displaced by solid

Density of liquid

= (W pl − W p)− (W pls − W ps)

ρ l

(1.30)where

V s= volume of the solid (m3),

W pl= weight of the pycnometer filled with liquid (kg),

W p= weight of the empty pycnometer (kg),

W pls= weight of the pycnometer containing the solid sample and filled with liquid (kg),

W ps= weight of the pycnometer containing solid sample with no liquid (kg),

ρ1= density of the liquid (kg/m3)

The volume of a sample can be measured by direct measurement of volume of the liquid displaced byusing a graduated cylinder or burette The difference between the initial volume of liquid in a graduatedcylinder and the volume of liquid with immersed material gives us the volume of the material That

is, the increase in volume after addition of solid sample is equal to the solid volume

In the liquid displacement method, liquids used should have a low surface tension and should beabsorbed very slowly by the particles Most commonly used fluids are water, alcohol, toluene, andtetrachloroethylene For displacement, it is better to use a nonwetting fluid such as mercury Coating

of a sample with a film or paint may be required to prevent liquid absorption

For larger objects, a platform scale can be used (Mohsenin, 1970) (Fig 1.8) The sample is pletely submerged in liquid such that it does not make contact with the sides or bottom of the beaker.Weight of the liquid displaced by the solid sample is divided by its density The method is based onthe Archimedes principle, which states that a body immersed in a fluid will experience a weight loss

com-in an amount equal to the weight of the fluid it displaces That is, the upward buoyancy force exerted

Figure 1.8 Platform scale for measurement of volume of large objects

18 1 Size, Shape, Volume, and Related Physical Attributes

on a body immersed in a liquid is equal to the weight of the displaced liquid

G= the buoyancy force (N),

ρ l= the density of liquid (kg/m3),

Wair= the weight of sample in air (kg),

W l = the weight of sample in liquid (kg)

Liquids having a density lower than that of sample should be used if partial floating of the sample isobserved The sample is forced into the liquid by means of a sinker rod if it is lighter or it is suspendedwith a string if it is heavier than the liquid If the sample is forced into the fluid using a sinker rod, itshould be taken into account in the measurement as:

V s =Gsample + sinker− Gsinker

ρ l

(1.32)

1.4.2 Gas Displacement Method

Volumes of particulate solids and materials with irregular shape can be determined by displacement

of gas or air in pycnometer (Karathanos & Saravacos, 1993) The most commonly used gases are

helium and nitrogen The pycnometer consists of two airtight chambers of nearly equal volumes, V1

and V2,that are connected with small-diameter tubing (Fig 1.9) The material to be measured is placed

in the second chamber The exhaust valve (valve 3) and the valve between the two chambers (valve2) are closed The inlet valve (valve 1) is opened and the gas is supplied to the first chamber until the

Pressure Gage

Valve 3 Valve 2

gauge pressure is increased up to a suitable value (e.g., 700–1000 Pa) Then, the inlet valve is closedand the equilibrium pressure is recorded Assuming that the gas behaves ideally:

where

P1= equilibrium pressure when valve 2 is closed (Pa),

V1= volume of the first chamber (m3),

n= moles of gas (kg mol),

R= gas constant (8314.34 J/kg mol K),

is not isothermal To eliminate these errors, the instrument should be calibrated by using an object ofprecisely known volume

1.4.3 Solid Displacement Method

The volume of irregular solids can also be measured by sand, glass bead, or seed displacementmethod Rapeseeds are commonly used for determination of volume of baked products such as bread

In the rapeseed method, first the bulk density of rapeseeds is determined by filling a glass container ofknown volume uniformly with rapeseeds through tapping and smoothing the surface with a ruler Allmeasurements are done until the constant weight is reached between the consecutive measurements.The densities of the seeds are calculated from the measured weight of the seeds and volume of thecontainer

Then, the sample and rapeseeds are placed together in the container The container is tapped andthe surface is smoothed with a ruler Tapping and smoothing are continued until a constant weight is

20 1 Size, Shape, Volume, and Related Physical Attributes

reached between three consecutive measurements The volume of the sample is calculated as follows:

pores that are filled with air It can be determined by the gas displacement method in which thegas is capable of penetrating all open pores up to the diameter of the gas molecule

(in-ternal pores) Apparent volume of regular geometries can be calculated using the characteristicdimensions Apparent volume of irregularly shaped samples may be determined by solid orliquid displacement methods

the pores enclosed within the material (internal pores) and also the void volume outside theboundary of individual particles when stacked in bulk (external pores)

For baked products, especially for cakes, sometimes an index of volume based on the dimensions

of the cake is used (Cloke, Davis, & Gordon, 1984) In this method, the cake is cut into two halves

A template is used to measure height from different positions of the cross section (Fig 1.10) Volumeindex determined by the AACC template method is based on the sum of height at different positions(AACC, 1983)

The bottom diameter ( A to E) was also measured and subtracted from the diameter of the baking

pan to obtain shrinkage value Uniformity, which is a measurement of cake symmetry, is found throughsubtraction of the two midpoint measurements:

Quality of food materials can be assessed by measuring their densities Density data of foodsare required in separation processes, such as centrifugation and sedimentation and in pneumatic andhydraulic transport of powders and particulates In addition, measuring the density of liquid is required

to determine the power required for pumping

A B C D E

Figure 1.10 Schematic cross-sectional tracing of a cake where, C is height at center and B and D are heights

at three fifths of distance from center to edge

Density can be calculated after measuring the mass and volume of the object because it is defined

as the mass per unit volume In the SI system, the unit of density is kg/m3

In most of the engineering problems, solids and liquids are assumed to be incompressible, that is, thedensity is hardly affected by moderate changes in temperature and pressure Gases are compressibleand their densities are affected by changes in temperature and pressure The densities of gases decrease

as temperature increases whereas they increase with increase in pressure Under moderate conditions,most gases obey the ideal gas law Molecular weight of any gas in kg (1 kg-mole) occupies 22.4 m3

at 273 K and 1 atm For example, density of air can be calculated from:

The density of liquid is calculated from the ratio of weight of the hydrometer to the volume of thedisplaced liquid:

where

W = weight of hydrometer (kg),

A= cross-sectional area of stem (m2),

X= the length of the stem immersed (m),

V = Volume of the bulb (m3)

22 1 Size, Shape, Volume, and Related Physical Attributes

as brix saccharometers for percentage of sucrose by weight in a solution, alcoholometers for percentage

of alcohol by volume, and salometers for determination of the percent saturation of salt solutions.The density of solids can be calculated from their measured weight and volume Volume measure-ment methods have been discussed in Section 1.4

Density can be expressed in different forms For example, for particulate materials such as grains, onemay be interested in the density of individual particles or the density of the bulk of the material whichincludes the void volume In literature, the definitions of densities differ form each other Therefore,the form of the density must be well defined before presenting the data The most commonly useddefinitions are:

densities of its components considering conservation of mass and volume If the densities and volume

or mass fractions of constituents are known, density can be determined from:



ρ i

(1.45)

that are filled with air It can be calculated by dividing the sample weight by solid volume determined

by the gas displacement method in which gas is capable of penetrating all open pores up to the diameter

of the gas molecule

broken into pieces small enough to be sure that no closed pores remain

the volume of all closed pores but not the externally connected ones It can be calculated by dividingthe sample weight by particle volume determined by a gas pycnometer

pores) Apparent density of regular geometries can be determined from the volume calculated usingthe characteristic dimensions and mass measured Apparent density of irregularly shaped samples may

be determined by solid or liquid displacement methods

particulate solids is measured by allowing the sample to pour into a container of known dimensions.Special care should be taken since the method of filling and the container dimensions can affect themeasurement It depends on the solid density, geometry, size, surface properties, and the method ofmeasurement It can be calculated by dividing the sample weight by bulk volume

The density of food materials depends on temperature and the temperature dependence of densities ofmajor food components [pure water, carbohydrate (CHO), protein, fat, ash and ice] has been presented

by Choi and Okos (1986) as follows:

where densities (ρ) are in kg/m3and temperatures (T ) are in◦C and varies between−40 and 150◦C.

Example 1.4 Calculate the true density of spinach at 20◦C having the composition given in TableE.1.4.1

24 1 Size, Shape, Volume, and Related Physical Attributes

Table E.1.4.1 Composition of Spinach

Water 91.57Protein 2.86Fat 0.35Carbohydrate 1.72Ash 3.50

Table E.1.4.2 Density and Mass Fraction (X w

The true density of spinach can be calculated using Eq (1.45):

Porosity (ε) is defined as the volume fraction of the air or the void fraction in the sample and

Figure 1.12 Image of bread sample (scale represents 1 cm).

1 Direct method: In this method, porosity is determined from the difference of bulk volume of apiece of porous material and its volume after destruction of all voids by means of compression Thismethod can be applied if the material is very soft and no attractive or repulsive force is present betweenthe particles of solid

2 Optical method: In this method, porosity is determined from the microscopic view of a section ofthe porous medium This method is suitable if the porosity is uniform throughout the sample, that is,the sectional porosity represents the porosity of whole sample Pore size distribution can be determined

if a suitable software is used to analyze images

Image J (http://rsb.info.nih.gov/ij/) is a software used to analyze the pores and to determine areabased pore size distribution, median pore diameter, and percent area fraction of pores This softwareuses the contrast between the two phases (pores and solid part) in the image (Abramoff, Magelhaes, &Ram, 2004) First, the image is obtained Then, the scanned color image is converted to gray scaleusing this software Using bars of known lengths, pixel values are converted into distance units.Figure 1.12 shows the image of a bread sample From the image, pore areas are extracted by thesoftware (Fig 1.13) The porosity based on area fraction for this bread sample is determined to be0.348 The area-based pore size distributions for the bread are shown in Fig 1.14

3 Density method: In this method, porosity is calculated from the measured densities

Porosity due to the enclosed air space within the particles is named apparent porosity (εapp) anddefined as the ratio of total enclosed air space or voids volume to the total volume It can also be namedinternal porosity Apparent porosity is calculated from the measured solid (ρ s) and apparent density(ρapp) data as:

εapp= 1 − ρapp

26 1 Size, Shape, Volume, and Related Physical Attributes

Figure 1.13 Extracted pores of bread using image J

Figure 1.14 Cumulative pore size distribution

Figure 1.15 Different kinds of pores.

or from the specific solid (V s ) and apparent (Vapp) volumes as:

εapp= 1 − V s

Bulk porosity (εbulk), which can also be called external or interparticle porosity, includes the voidvolume outside the boundary of individual particles when stacked as bulk and calculated using bulkand apparent densities as:

Pores within the food materials (internal pores) can be divided into three groups: closed pores thatare closed from all sides, blind pores that have one end closed, and open or flow- through pores wherethe flow typically takes place (Fig 1.15)

Since the apparent porosity is due to the enclosed air space within the particles and there are threedifferent forms of pores within the particles, it can be written as:

where

εCP= porosity due to closed pores,

εOP = porosity due to open or flow through pores,

εBP= porosity due to blind pores

Then, total porosity can also be written as: